-- |
-- Module:      Math.NumberTheory.MoebiusInversion
-- Copyright:   (c) 2012 Daniel Fischer
-- Licence:     MIT
-- Maintainer:  Daniel Fischer <daniel.is.fischer@googlemail.com>
--
-- Generalised Möbius inversion

{-# LANGUAGE BangPatterns        #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE ScopedTypeVariables #-}

module Math.NumberTheory.MoebiusInversion
    ( generalInversion
    , totientSum
    ) where

import Control.Monad
import Control.Monad.ST
import Data.Proxy
import qualified Data.Vector.Generic as G
import qualified Data.Vector.Generic.Mutable as MG

import Math.NumberTheory.Roots
import Math.NumberTheory.Utils.FromIntegral

-- | @totientSum n@ is, for @n > 0@, the sum of @[totient k | k <- [1 .. n]]@,
--   computed via generalised Möbius inversion.
--   See <http://mathworld.wolfram.com/TotientSummatoryFunction.html> for the
--   formula used for @totientSum@.
--
-- >>> import Data.Proxy
-- >>> totientSum (Proxy :: Proxy Data.Vector.Unboxed.Vector) 100 :: Int
-- 3044
-- >>> totientSum (Proxy :: Proxy Data.Vector.Vector) 100 :: Integer
-- 3044
totientSum
    :: (Integral t, G.Vector v t)
    => Proxy v
    -> Word
    -> t
totientSum :: forall t (v :: * -> *).
(Integral t, Vector v t) =>
Proxy v -> Word -> t
totientSum Proxy v
_ Word
0 = t
0
totientSum Proxy v
proxy Word
n = Proxy v -> (Word -> t) -> Word -> t
forall t (v :: * -> *).
(Num t, Vector v t) =>
Proxy v -> (Word -> t) -> Word -> t
generalInversion Proxy v
proxy (t -> t
forall {a}. Integral a => a -> a
triangle (t -> t) -> (Word -> t) -> Word -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Word -> t
forall a b. (Integral a, Num b) => a -> b
fromIntegral) Word
n
  where
    triangle :: a -> a
triangle a
k = (a
k a -> a -> a
forall a. Num a => a -> a -> a
* (a
k a -> a -> a
forall a. Num a => a -> a -> a
+ a
1)) a -> a -> a
forall a. Integral a => a -> a -> a
`quot` a
2

-- | @generalInversion g n@ evaluates the generalised Möbius inversion of @g@
--   at the argument @n@.
--
--   The generalised Möbius inversion implemented here allows an efficient
--   calculation of isolated values of the function @f : N -> Z@ if the function
--   @g@ defined by
--
-- >
-- > g n = sum [f (n `quot` k) | k <- [1 .. n]]
-- >
--
--   can be cheaply computed. By the generalised Möbius inversion formula, then
--
-- >
-- > f n = sum [moebius k * g (n `quot` k) | k <- [1 .. n]]
-- >
--
--   which allows the computation in /O/(n) steps, if the values of the
--   Möbius function are known. A slightly different formula, used here,
--   does not need the values of the Möbius function and allows the
--   computation in /O/(n^0.75) steps, using /O/(n^0.5) memory.
--
--   An example of a pair of such functions where the inversion allows a
--   more efficient computation than the direct approach is
--
-- >
-- > f n = number of reduced proper fractions with denominator <= n
-- >
-- > g n = number of proper fractions with denominator <= n
-- >
--
--   (a /proper fraction/ is a fraction @0 < n/d < 1@). Then @f n@ is the
--   cardinality of the Farey sequence of order @n@ (minus 1 or 2 if 0 and/or
--   1 are included in the Farey sequence), or the sum of the totients of
--   the numbers @2 <= k <= n@. @f n@ is not easily directly computable,
--   but then @g n = n*(n-1)/2@ is very easy to compute, and hence the inversion
--   gives an efficient method of computing @f n@.
--
--   Since the function arguments are used as array indices, the domain of
--   @f@ is restricted to 'Int'.
--
--   The value @f n@ is then computed by @generalInversion g n@. Note that when
--   many values of @f@ are needed, there are far more efficient methods, this
--   method is only appropriate to compute isolated values of @f@.
generalInversion
    :: (Num t, G.Vector v t)
    => Proxy v
    -> (Word -> t)
    -> Word
    -> t
generalInversion :: forall t (v :: * -> *).
(Num t, Vector v t) =>
Proxy v -> (Word -> t) -> Word -> t
generalInversion Proxy v
proxy Word -> t
fun Word
n = case Word
n of
    Word
0 ->[Char] -> t
forall a. HasCallStack => [Char] -> a
error [Char]
"Möbius inversion only defined on positive domain"
    Word
1 -> Word -> t
fun Word
1
    Word
2 -> Word -> t
fun Word
2 t -> t -> t
forall a. Num a => a -> a -> a
- Word -> t
fun Word
1
    Word
3 -> Word -> t
fun Word
3 t -> t -> t
forall a. Num a => a -> a -> a
- t
2t -> t -> t
forall a. Num a => a -> a -> a
*Word -> t
fun Word
1
    Word
_ -> (forall s. ST s t) -> t
forall a. (forall s. ST s a) -> a
runST (Proxy v -> (Int -> t) -> Int -> ST s t
forall s t (v :: * -> *).
(Num t, Vector v t) =>
Proxy v -> (Int -> t) -> Int -> ST s t
fastInvertST Proxy v
proxy (Word -> t
fun (Word -> t) -> (Int -> Word) -> Int -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Word
intToWord) (Word -> Int
wordToInt Word
n))

fastInvertST
    :: forall s t v.
       (Num t, G.Vector v t)
    => Proxy v
    -> (Int -> t)
    -> Int
    -> ST s t
fastInvertST :: forall s t (v :: * -> *).
(Num t, Vector v t) =>
Proxy v -> (Int -> t) -> Int -> ST s t
fastInvertST Proxy v
_ Int -> t
fun Int
n = do
    let !k0 :: Int
k0 = Int -> Int
forall {a}. Integral a => a -> a
integerSquareRoot (Int
n Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
2)
        !mk0 :: Int
mk0 = Int
n Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
k0Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
        kmax :: a -> a -> a
kmax a
a a
m = (a
a a -> a -> a
forall a. Integral a => a -> a -> a
`quot` a
m a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Integral a => a -> a -> a
`quot` a
2

    small <- Int -> ST s (Mutable v (PrimState (ST s)) t)
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
Int -> m (v (PrimState m) a)
MG.unsafeNew (Int
mk0 Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
1) :: ST s (G.Mutable v s t)
    MG.unsafeWrite small 0 0
    MG.unsafeWrite small 1 $! fun 1
    when (mk0 >= 2) $
        MG.unsafeWrite small 2 $! (fun 2 - fun 1)

    let calcit :: Int -> Int -> Int -> ST s (Int, Int)
        calcit Int
switch Int
change Int
i
            | Int
mk0 Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
i   = (Int, Int) -> ST s (Int, Int)
forall a. a -> ST s a
forall (m :: * -> *) a. Monad m => a -> m a
return (Int
switch,Int
change)
            | Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
change = Int -> Int -> Int -> ST s (Int, Int)
calcit (Int
switchInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) (Int
change Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
4Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
switchInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
6) Int
i
            | Bool
otherwise = do
                let mloop :: t -> Int -> Int -> ST s (Int, Int)
mloop !t
acc Int
k !Int
m
                        | Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
switch    = t -> Int -> ST s (Int, Int)
kloop t
acc Int
k
                        | Bool
otherwise     = do
                            val <- Mutable v (PrimState (ST s)) t -> Int -> ST s t
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> m a
MG.unsafeRead Mutable v s t
Mutable v (PrimState (ST s)) t
small Int
m
                            let nxtk = Int -> Int -> Int
forall a. Integral a => a -> a -> a
kmax Int
i (Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
                            mloop (acc - fromIntegral (k-nxtk)*val) nxtk (m+1)
                    kloop :: t -> Int -> ST s (Int, Int)
kloop !t
acc Int
k
                        | Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0    = do
                            Mutable v (PrimState (ST s)) t -> Int -> t -> ST s ()
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> a -> m ()
MG.unsafeWrite Mutable v s t
Mutable v (PrimState (ST s)) t
small Int
i (t -> ST s ()) -> t -> ST s ()
forall a b. (a -> b) -> a -> b
$! t
acc
                            Int -> Int -> Int -> ST s (Int, Int)
calcit Int
switch Int
change (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
                        | Bool
otherwise = do
                            val <- Mutable v (PrimState (ST s)) t -> Int -> ST s t
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> m a
MG.unsafeRead Mutable v s t
Mutable v (PrimState (ST s)) t
small (Int
i Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1))
                            kloop (acc-val) (k-1)
                t -> Int -> Int -> ST s (Int, Int)
mloop (Int -> t
fun Int
i t -> t -> t
forall a. Num a => a -> a -> a
- Int -> t
fun (Int
i Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
2)) ((Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
2) Int
1

    (sw, ch) <- calcit 1 8 3
    large <- MG.unsafeNew k0 :: ST s (G.Mutable v s t)

    let calcbig :: Int -> Int -> Int -> ST s (G.Mutable v s t)
        calcbig Int
switch Int
change Int
j
            | Int
j Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0    = Mutable v s t -> ST s (Mutable v s t)
forall a. a -> ST s a
forall (m :: * -> *) a. Monad m => a -> m a
return Mutable v s t
large
            | (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
change Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
n   = Int -> Int -> Int -> ST s (Mutable v s t)
calcbig (Int
switchInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) (Int
change Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
4Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
switchInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
6) Int
j
            | Bool
otherwise = do
                let i :: Int
i = Int
n Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)
                    mloop :: t -> Int -> Int -> ST s (Mutable v s t)
mloop !t
acc Int
k Int
m
                        | Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
switch    = t -> Int -> ST s (Mutable v s t)
kloop t
acc Int
k
                        | Bool
otherwise     = do
                            val <- Mutable v (PrimState (ST s)) t -> Int -> ST s t
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> m a
MG.unsafeRead Mutable v s t
Mutable v (PrimState (ST s)) t
small Int
m
                            let nxtk = Int -> Int -> Int
forall a. Integral a => a -> a -> a
kmax Int
i (Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
                            mloop (acc - fromIntegral (k-nxtk)*val) nxtk (m+1)
                    kloop :: t -> Int -> ST s (Mutable v s t)
kloop !t
acc Int
k
                        | Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0    = do
                            Mutable v (PrimState (ST s)) t -> Int -> t -> ST s ()
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> a -> m ()
MG.unsafeWrite Mutable v s t
Mutable v (PrimState (ST s)) t
large (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) (t -> ST s ()) -> t -> ST s ()
forall a b. (a -> b) -> a -> b
$! t
acc
                            Int -> Int -> Int -> ST s (Mutable v s t)
calcbig Int
switch Int
change (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)
                        | Bool
otherwise = do
                            let m :: Int
m = Int
i Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` (Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
                            val <- if Int
m Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
mk0
                                     then Mutable v (PrimState (ST s)) t -> Int -> ST s t
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> m a
MG.unsafeRead Mutable v s t
Mutable v (PrimState (ST s)) t
small Int
m
                                     else Mutable v (PrimState (ST s)) t -> Int -> ST s t
forall (m :: * -> *) (v :: * -> * -> *) a.
(PrimMonad m, MVector v a) =>
v (PrimState m) a -> Int -> m a
MG.unsafeRead Mutable v s t
Mutable v (PrimState (ST s)) t
large (Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
*(Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)
                            kloop (acc-val) (k-1)
                t -> Int -> Int -> ST s (Mutable v s t)
mloop (Int -> t
fun Int
i t -> t -> t
forall a. Num a => a -> a -> a
- Int -> t
fun (Int
i Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
2)) ((Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
2) Int
1

    mvec <- calcbig sw ch k0
    MG.unsafeRead mvec 0